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In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle. The hyperbolic angle is first defined for a "standard position", and subsequently as a measure of an interval on a branch of a hyperbola. A hyperbolic angle in ''standard position'' is the angle at (0, 0) between the ray to (1, 1) and the ray to (''x'', 1/''x'') where ''x'' > 1. The ''magnitude'' of the hyperbolic angle is the area of the corresponding hyperbolic sector which is ln ''x''. Note that unlike circular angle, hyperbolic angle is ''unbounded'', as is the function ln ''x'', a fact related to the unbounded nature of the harmonic series. The hyperbolic angle in standard position is considered to be negative when 0 < ''x'' < 1. Suppose ''ab'' = 1 and ''cd'' = 1 with ''c'' > ''a'' > 1 so that (''a'', ''b'') and (''c'', ''d'') determine an interval on the hyperbola ''xy'' = 1. Then the squeeze mapping with diagonal elements ''b'' and ''a'' maps this interval to the standard position hyperbolic angle that runs from (1, 1) to (''bc'', ''ad''). By the result of Gregoire de Saint-Vincent, the hyperbolic sector determined by (''a'', ''b'') and (''c'', ''d'') has the same area as this standard position angle, and the magnitude of the hyperbolic angle is taken to be this area. The hyperbolic functions sinh, cosh, and tanh use the ''hyperbolic angle'' as their independent variable because their values may be premised on analogies to circular trigonometric functions when the hyperbolic angle defines a hyperbolic triangle. Thus this parameter becomes one of the most useful in the calculus of a real variable. ==Comparison with circular angle== In terms of area, one can consider a circle of radius √2 for which the area of a circular sector of ''u'' radians is . (The area of the whole circle is 2π.) As the hyperbola , associated with the hyperbolic angle, has shortest diameter between and , it too has semidiameter √2. As shown in the diagram, a ray of slope less than one determines an angle ''u'' that is a circular angle of magnitude equal to a circular sector, or a hyperbolic angle. The circular and hyperbolic trigonometric function magnitudes are all √2 times the legs of right triangles determined by the ray, circle, and hyperbola. There is also a projective resolution between circular and hyperbolic cases: both curves are conic sections, and hence are treated as projective ranges in projective geometry. Given an origin point on one of these ranges, other points correspond to angles. The idea of addition of angles, basic to science, corresponds to addition of points on one of these ranges as follows: Circular angles can be characterised geometrically by the property that the if two chords ''P''0''P''1 and ''P''0''P''2 subtend angles ''L''1 and ''L''2 at the centre of a circle, their sum is the angle subtended by a chord ''PQ'', where ''PQ'' is required to be parallel to ''P''1''P''2. The same construction can also be applied to the hyperbola. If ''P''0 is taken to be the point , ''P''1 the point , and ''P''2 the point , then the parallel condition requires that ''Q'' be the point . It thus makes sense to define the hyperbolic angle from ''P''0 to an arbitrary point on the curve as a logarithmic function of the point's value of ''x''.〔Bjørn Felsager, (Through the Looking Glass – A glimpse of Euclid's twin geometry, the Minkowski geometry ), ICME-10 Copenhagen 2004; p.14. See also example sheets () () exploring Minkowskian parallels of some standard Euclidean results〕〔Viktor Prasolov and Yuri Solovyev (1997) ''Elliptic Functions and Elliptic Integrals'', page 1, Translations of Mathematical Monographs volume 170, American Mathematical Society〕 Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a pseudo-Euclidean plane steadily moving orthogonally to a ray from the origin traces out a hyperbola. In Euclidean space, the multiple of a given angle traces equal distances around a circle while it traces exponential distances upon the hyperbolic line.〔(Hyperbolic Geometry ) pp 5–6, Fig 15.1〕 Both circular and hyperbolic angle provide instances of an invariant measure. Arcs with an angular magnitude on a circle generate a measure on certain measurable sets on the circle whose magnitude does not vary as the circle turns or rotates. For the hyperbola the turning is by squeeze mapping, and the hyperbolic angle magnitudes stay the same when the plane is squeezed by a mapping :(''x'', ''y'') ↦ (''rx'', ''y'' / ''r''), with ''r'' > 0 . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「hyperbolic angle」の詳細全文を読む スポンサード リンク
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